I work at the interface of algebraic geometry and commutative algebra. I enjoy thinking about non-Noetherian objects that arise in geometry and arithmetic (such as valuation rings and perfectoid spaces), and their interactions with varieties and Noetherian rings. Here is a more detailed description of my research.

Notes

Notes are subject to change without notice.

Notes on Huber rings for a learning seminar at the University of Michigan (Winter 2017). Last updated Feb 18, 2017. A new section was added on uniform Huber rings (not discussed in the lectures), following Bhargav’s discussion of uniform K-Banach algebras in his course. In particular, we prove equivalence of categories results generalizing [Bha17, Thm 9.7 and Cor 9.9].

On a vanishing result in sheaf cohomology. An example is given of a non-quasicompact scheme that violates a vanishing result in sheaf cohomology that holds for certain quasicompact spaces [Stacks Project, Tag02UX]. This example can be interpreted purely topologically (without mentioning schemes), and is incorporated in the latter form in Tag0BX0.

Notes from a summer mini-course I taught at Michigan on notions of singularities in prime characteristic in 2016. The notes have not been proof-read and do not cover a lot of material.